Is there a simple way to fix this, or am I using the wrong approach entirely here?
That's never going to look exactly right unless you draw each separately and tweak the corner radius manually, which would be a right pain in the arse.
Fortunately, there's another way.
Draw your rounded rectangle. Increase the stroke weight until the thickness of the stroke is about the distance you want between your double outline.
Then go Object -> Outline Stroke.
Ungroup the result with CTRL-SHIFT-G and you have two separate shapes, one is the original fill for the rectangle, but your stroke is now a double path. Change the fill to whatever colour you want the gap between the double stroke to have, and then add a stroke colour.